Introduction to lie algebras and representation theory

Determine the value of the variable in each equation.

Basic Algebra = 代数学入門

The category of all 2-dimensional real division algebras is shown to split into four full subcategories, each of which is given by the natural action of a Coxeter group of type &#x1d538;1 or &#x1d538;2 on the set of all pairs of ellipses in ℝ2, which are centred in the origin and have reciprocal axis lengths. Cross-sections for the orbit sets of these group actions are being determined. They yield a classification of all 2-dimensional real division algebras. Moreover all morphisms between the objects in this classifying list are described, and thus an explicit and geometric picture of the category of all 2-dimensional real division algebras is obtained. This elementary and self-contained exposition extends Darpo and Dieterich's recent description [14] of the category of all 2-dimensional commutative real division algebras, which in turn is based on Benkart, Britten and Osborn's investigation [4] of the isotopes of ℂ. It also supplements earlier contributions of Althoen and Kugler [2], Burdujan [9], Gottschling [25], Petersson [33], Hubner and Petersson [29], and Dokovic and Zhao [23] to the problem of classifying all 2-dimensional real division algebras.

Classification, Automorphism Groups and Categorical Structure of the Two-Dimensional Real Division Algebras

http://icm2006.org/v_f/AbsDef/Posters/abs_1490.pdf

Ternary derivations of finite-dimensional real division algebras

On the classification of four-dimensional quadratic division algebras over square-ordered fields

On the Classification of 4‐Dimensional Quadratic Division Algebras Over Square‐Ordered Fields

Zu jeder Konfiguration des IR3, welche aus einem Ellipsoid in Hauptachsenform und einem Punktepaar besteht, wird eine vierdimensionale reelle Divisionsalgebra konstruiert. Diese Konstruktion induziert eine Bijektion zwischen den Aquivalenzklassen von Konfigurationen und den Isomorphieklassen vierdimensionaler potenzassoziativer reeller Divisionsalgebren. Hierbei sind zwei Konfigurationen genau dann aquivalent, wenn ihre Ellipsoide ubereinstimmen und ihre Punktepaare in derselben Bahn unter der Operation der eigentlichen Symmetriegruppe des Ellipsoids liegen. Ein Querschnitt fur die Aquivalenzklassen von Konfigurationen wird angegeben.

Zur Klassifikation Vierdimensionaler Reeller Divisionsalgebren

An absolute valued algebra is a non-zero real algebra that is equipped with a multiplicative norm. We classify all finite dimensional absolute valued algebras having a non-zero central idempotent or a one-sided unity, up to algebra isomorphism. This completes earlier results of Ramirez Alvarez and Rochdi which, in our self-contained presentation, are recovered from the wider context of composition k-algebras with an LR-bijective idempotent.

Classification of the finite dimensional absolute valued algebras having a non-zero central idempotent or a one-sided unity

/pdf/quadratic-division-algebras-revisited-remarks-on-an-article-59eeks8kcf.pdf

Ernst Dieterich

Papers

Classification, Automorphism Groups and Categorical Structure of the Two-Dimensional Real Division Algebras

On the Classification of 4‐Dimensional Quadratic Division Algebras Over Square‐Ordered Fields

Zur Klassifikation Vierdimensionaler Reeller Divisionsalgebren

Classification of the finite dimensional absolute valued algebras having a non-zero central idempotent or a one-sided unity

Quadratic division algebras revisited. (Remarks on an article by J. M. Osborn.)